Method for production planning in an uncertain demand environment

ABSTRACT

A decision-making method suitable for production planning in an uncertain demand environment. To this end, the method comprises combining an implosion technology with a scenario-based analysis, thus manifesting, a sui generis capability which preserves the advantages and benefits of each of its subsumed aspects.

FIELD OF THE INVENTION

This invention relates to a decision-making method suitable forproduction planning in an uncertain demand environment.

INTRODUCTION TO THE INVENTION

One of the important issues in production planning is to deal withuncertainties associated with demand. While a vast volume of theliterature addresses this issue, production/materials planning underuncertainty still presents a significant challenge to researchers andpractitioners.

There have been many interesting studies in inventory/productionplanning theory. However, we have discerned that the results obtained sofar are either based on over-simplified assumptions, or arecomputationally intractable for real-world problems.

For example, the well-known EOQ (Economic Order Quantity) model inventedin the early part of this century is considered to be one of theearliest and most important developments in the mathematicalinventory/production planning theory. The basic EOQ model still remainsthe most widely used analytical method for inventory control inpractice. In the 50's and 60's, more serious mathematical analyses ofvarious inventory problems were undertaken and became the fundamentalsof later developments in the area.

The solutions available in the literature can be categorized into twogeneral types. One type of solution can be considered to be theextensions of the EOQ model where a simple form solution can be obtainedunder certain assumptions about demand, costs, and other parameters.Especially, the studies on single product and single location problemshave produced a rich collection of analytical models, including manyvariations of the EOQ model and (s,S) models. This type of solution isusually easy to implement and interpret because of its simplicity.However, the assumptions made to ensure the validity of the solution areoften restrictive and may not be consistent with the reality. Anothergeneral type of solution normally involves a mathematical programmingformulation, which allows more flexible modeling of the underlyingproduction/inventory process, and, therefore, can handle a wide range ofreal-world applications. A major limitation to the use of mathprogramming solutions is often the large size of real-world problems,even though the advances over the years of computer hardware andsoftware have greatly enhanced our ability to solve large scale mathprogramming problems.

In the last two decades, material requirements planning (MRP) has becomea common practice in industry for the purposes of production planningand control. The earlier versions of MRP largely focused on managingmaterials. The same concepts were applied to labor, another importantfactor in planning. Beyond labor and material, further applicationsdealt with equipment, tooling and other resources. These variations gaverise to the broader term, manufacturing resource planning, which isoften referred to as MRP II.

As a particular example, we note the implosion(TM) technology developedin IBM which is able to provide feasible and optimal production plansunder materials and capacity constraints. Traditional MRP systemsperform requirements analysis by using demands and the Bill Of Materials(BOM) to determine the necessary resources to fulfill the demand. Incontrast, implosion(TM)-based systems can perform resource allocationunder constraints by using demands, available resources, and the Bill OfManufacture (includes BOM as well as Bill of Capacities) to determine afeasible product mix which meets the user goals. These goals correspondto user defined criteria such as customer serviceability, profitmaximization, inventory minimization, and revenue maximization.

However, the effectiveness of MRP-based systems may be limited by theweaknesses of the basic MRP framework. Particularly, uncertainty isignored in the standard MRP approach. All parameters, such as the futuredemand, production rates, yields, lead times, etc., are treated as ifthey were known with certainty.

One useful technique often used to deal with the uncertainty is thescenario-based analysis. Scenarios are usually used as a simplified wayof representing the uncertainty. By performing multiple optimizationruns against different scenarios, one can gain the insights needed toplan effectively for an uncertain future. Escudero and Kamesam (1992)provide a scenario-based solution methodology for solving aggregateproduction planning problems under demand uncertainty. They obtain animplementable policy by solving a stochastic LP problem. (See Escudero,L. and P. Kamersom, MRP Modeling via Scenarios, Research Report,RC-17982.)

SUMMARY OF THE INVENTION

We have now discovered a "Payoff Table" approach that is designed as adecision-making methodology for production planning in an uncertaindemand environment. The methodology comprises combining the power of thescenario-based analysis and the implosion technology. For eachindividual demand scenario, the implosion method may be used to providea deterministic solution which is optimal given the particular demandscenario. Furthermore, we also compute the performance measure of eachsolution against all other potential demand scenarios. The completeenumeration of performance measures for each solution against all demandscenarios produces a payoff table, which may be referred to as aproduction plan payoff table (PPPT).

Accordingly, we disclose a novel program storage device readable by amachine, tangibly embodying a program of instructions executable by themachine to perform method steps for production planning in an uncertaindemand environment, said method steps comprising:

1) representing the uncertainty in the demand environment by employing ascenario-based analysis including the steps of performing multipleoptimization runs against different demand scenarios; and

2) combining an implosion technology with the scenario-based analysisfor generating for any one individual demand scenario, a deterministicsolution which is optimal for the particular demand scenario.

The novel method as defined can realize important advantages, asenumerated below. Here, it is significant to note that the method, byway of its agency of combining implosion technology with ascenario-based analysis, can perserve the benefits of each disparateaspect, while manifesting in combination a sui generis capability ofqualitative advantage and utility over the prior art.

BRIEF DESCRIPTION OF THE DRAWING

The invention is illustrated in the accompanying drawing, in which:

FIG. 1 shows a canonical scenario tree;

FIG. 2 shows a general scenario tree; and

FIGS. 3-8 show illustrative output computer display windows generated inaccordance with the present invention.

DETAILED DESCRIPTION OF THE INVENTION

The detailed description of the invention is organized as follows. InSection 1, we introduce the notation and present a deterministic versionof materials planning problems. The concept of Payoff Table is discussedin Section 2. Illustrative algorithms are provided in Section 3. Wedescribe the graphical user interface developed for Payoff Table inSection 4. Finally, a complete example is presented in Section 5.

1. Formulation of the Materials Planning Problem

To formulate the problem defined in the PPPT computation, we introducethe following notation for a deterministic materials planning problem.

T, set of periods that comprise the planning horizon.

J, set of products.

J_(e) .OR right.J, set of end products.

J_(a) .OR right.J, set of subassemblies.

I, set of components.

I_(r) .OR right.I, set of out-sourcing components. (Note, I_(r) ∩J_(a)=empty and I=I_(r) ∪J_(a).)

I_(j) .OR right.I, set of components in the BOM of product j.di-electcons.J.

d_(j),t, demand for end product j.di-elect cons.J_(e) in period t.

c_(j), cycle time of product j.di-elect cons.J.

O_(ij) offset for component i in the cycle time of product j fori.di-elect cons.I_(j) and j.di-elect cons.J.

a_(ij) amount of component i that is needed by one unit of product j fori.di-elect cons.I_(j) j and j.di-elect cons.J.

h_(i),j, unit holding cost for component i.di-elect cons.I in period t.

h^(c) _(j),t, unit holding cost for end product j.di-elect cons.J_(c) inperiod t.

r_(j),t, unit penalty for unsatisfied demand of end product j.di-electcons.J_(e) in period t.

Variables:

C_(j),t, ending inventory of product j.di-elect cons.J_(e) in period t.

Z_(j),t, production volume of end product j.di-elect cons.J_(e) inperiod t.

L_(j),t, unsatisfied demand of end product j.di-elect cons.J_(e) inperiod t.

Q_(i),t, ending inventory of component i.di-elect cons.I in period t.

Y_(i),t, consumed volume of component i.di-elect cons.I in period t.

X_(i),t, production/procurement volume of component i.di-elect cons.I inperiod t.

Note

C_(j),0, initial inventory of product j.di-elect cons.J_(e) at thebeginning of the planning horizon.

Q_(i),0, initial inventory of component i.di-elect cons.I at thebeginning of the planning horizon.

Production and procurement decisions are made at the beginning of eachperiod based on the information available at that time. Demandsmaterialize by the end of each period. Unsatisfied demand will bebacklogged, and inventories will be carried over to the next period. Atthe end of the planning horizon, all left over inventories of componentsand end products will be savaged.

1.1. Deterministic

A Linear Program can be formulated as follows. ##EQU1## subject to

    C.sub.j,t-1 +Z.sub.j,t +L.sub.j,t =d.sub.j,t +C.sub.j,t, t.di-elect cons.T,j.di-elect cons.J.sub.e

    Q.sub.i,t-1 +X.sub.i,t -Y.sub.i,t =Q.sub.i,t, t.di-elect cons.T, i.di-elect cons.I ##EQU2##

    C.sub.j,t,Q.sub.i,t, Z.sub.j,t, X.sub.i,t, Y.sub.i,t ≧0, 0≦L.sub.j,t ≦d.sub.j,t t.di-elect cons.T,j.di-elect cons.J.sub.e,i.di-elect cons.I

where ##EQU3##

The LP formulation presented above is a simplified version of a typicalmaterials planning problem with deterministic demands. In thisformulation, all the decisions are made at the beginning of the planninghorizon. The solution of this problem can be obtained using an implosiontechnology-based optimization engine, such as Supply Capability Engine(SCE). For the sake of simplicity, many advanced features that can behandled by implosion technology are omitted in this formulation.

Notice that the production cost and the procurement cost are notincluded in this formulation. However, it is easy to show that whenthese costs are linear and time-invariant, they do not affect thesolution of the optimization problem. Furthermore, the cost minimizationformulation presented here is equivalent to a profit maximizationformulation since the demand, hence the revenue from sales, isindependent of the production decision. Even in the case of lost sales,the situation can be handled by including the loss in sales as a penaltycost of the unsatisfied demand.

The formulation presented here describes a single-scenario problem. Whenthe demand uncertainty is modeled via scenarios, the above formulationcan still be used to obtain a solution for each scenario individually.However, the single-scenario solution may perform badly when a differentscenario actually occurs.

1.2. Scenario-based analysis

Let

S=set of scenarios.

N=number of the scenarios.

D^(s) =demand under scenario s.di-elect cons.S. D^(s) can be expressedby the matrix ##EQU4## where n is the number of periods in the planninghorizon, m is the number of the products. To simplify the notation, wesuppress the superscript for each element in the matrix.

P^(s) =production decision under scenario s.di-elect cons.S. It isreferred to as the Scenario solution for scenario s.

Note that a scenario solution consists of both a production schedule forall products and a procurement schedule for all components in eachperiod of the entire planning horizon. It can be expressed by thefollowing matrix ##EQU5## where l is the number of components.

To deal with the demand uncertainty over a period of time, recourseactions may be taken so that unimplemented decisions can be modifiedaccording to new information when it becomes available. For example, att=1, Z₁₁ is implemented; Z₁₂, . . . , Z_(ln) are computed but notimplemented. There are different types of recourse actions that can betaken. We will discuss two possibilities: the simple recourse and thefull recourse.

In the simple recourse, production decisions cannot be changed (even asnew information becomes available). In this case, the LP formulation issimilar to that of the single-scenario case. Nevertheless, the objectivefunction will be the weighted-average of the objective functions forindividual scenarios, and all the constraints have to be duplicated foreach scenario.

The full recourse allows all the production and procurement decisions tobe revisited every time period and adjustments can be made based on thelatest information. With full recourse, additional constraintsreflecting the nonanticipativity assumption of a production decisionmust be added in the model. The definition of the nonanticipativity isgiven in the next section.

To obtain an optimal solution of a multi-scenario problem usuallyrequires solving a stochastic LP program. Especially in the fullrecourse case it is far more complex than that for the single-scenariomodel. Therefore, a heuristic-based solution like the payoff tableapproach becomes necessary.

The payoff table approach is also a useful tool for sensitivityanalysis. We can use the payoff table to find out the expectedperformance of a particular production plan under different demandscenarios.

2. The concept of PPPT

The PPPT is a tool for production planning decision-making based onscenario analysis and the IBM implosion technology.

The PPPT computation is based on the following key concepts.

Scenarios are used to represent possible realizations of uncertaindemand.

For each demand scenario, a deterministic solution approach (such as theIBM implosion technology) can be used to produce a scenario-dependentproduction plan. It is clear that a production plan based on aparticular demand scenario is optimal only if the actual demand scenarioturns out to be the same scenario used for the planning.

In reality a different demand scenario may actually occur, andtherefore, the production plan may not be optimal for the actual demandscenario. To minimize the unfavorable impact of the mismatch between theproduction plan and the actual demand scenario, we would like toevaluate the expected overall performance measure and the robustness ofa production plan against all different demand scenarios.

Furthermore, when a production plan is evaluated against a differentdemand scenario, we need to keep in mind that the production plan willbe re-optimized when new information about demand becomes available, andonly the initial portion of the production plan has to be fixed andimplemented.

Based on the evaluation for each scenario-dependent production plan, wewill be able to choose one based on the expected performance or therobustness of the production plan against all possible demand scenarios.

2.1. Scenario Representation for Demand

The tree structure is utilized internally to represent demand scenarios.A general scenario tree can be illustrated by FIG. 1 (numeral 12).

Each node except the root represents the demands for all products in agiven period. A complete path from the root to an end node forms ademand scenario. Different demand scenarios may have the same demandsinitially and then diverge from a certain point. A special type ofscenario tree is the ones with the canonical structure. In a canonicaltree, the root is the only common node for any two branches. A canonicaldemand scenario tree means that all demand scenarios diverge from thefirst period in the planning horizon.

The scenario table is provided by users as an input. It specifies thedemand scenarios over the planning horizon. Suppose there are N demandscenarios and the number of periods in the planning horizon is n. Ascenario table is an N×(n+1) matrix. Each row of the scenario tabledescribes a demand scenario with the first n elements representing thedemands in the n periods and the last element being the probability ofthat the scenario will occur. A demand for a given scenario in a givenperiod is labeled by an integer. Usually, the first n elements ofScenario 1 are assigned to be 1's. For Scenario 2, if the demand in agiven period is different from the that in the same period for Scenario1, then the number 2 will be used to represent the demand for Scenario1; if the demand is the same with that in the same period for Scenario1, the number will remain the same. The same procedure applies for therest of the scenarios as well. Notice that demands in different periodscan also be represented by the same integer number. But the actualdemands can be different in different periods. In fact, the actualdemand quantities will be provided as a separate input by users.

In the following example, we have a problem with 3 different scenarios.The planning horizon is 2 periods. The scenario table is given below:

1 1 0.5

1 2 0.3

2 3 0.2

The corresponding scenario tree is shown in FIG. 2 (numeral 14).

2.2. The structure of PPPT

In the PPPT computation, the scenario-based representation for demand isused. A demand scenario is a multi-period statement of demand for agroup of products. A set of demand scenarios and the probabilitiesassociated with each scenarios are provided as inputs to represent theuncertain demand.

For each of the demand scenarios, the PPPT first computes the optimalproduction plan under a certain performance criterion. For a givendemand scenario, the optimal production plan specifies the productionquantities for each product in each period with the best overallperformance measure under materials and capacity constraints. Then theperformance measures of the optimal production plan for the given demandscenario are computed against all other demand scenarios. A completepayoff table is constructed by repeating this process for all the demandscenarios. The structure of a payoff table is illustrated in Table 1.

                  TABLE 1                                                         ______________________________________                                        The structure of the Payoff Table                                                     scenario                                                              ______________________________________                                        initial plan                                                                            D.sup.1                                                                              D.sup.2                                                                              . . .                                                                              D.sup.N                                                                            E    Δ.sup.+                                                                      Δ.sup.-                     P.sup.1   R.sub.1,1                                                                            R.sub.1,2                                                                            . . .                                                                              R.sub.1,N                                                                          E.sub.1                                                                            Δ.sub.1 .sup.+                                                               Δ.sub.1 .sup.-              P.sup.2                                                                       .                                                                             .                                                                             P.sup.N   R.sub.N,1                                                                            R.sub.N,2                                                                            . . .                                                                              R.sub.N,N                                                                          E.sub.N                                                                            Δ.sub.N .sup.+                                                               Δ.sub.N .sup.-              ______________________________________                                    

The interpretation of the elements in the table is given below.

R_(i).i --the optimal payoff for scenario i;

R_(ij) --the payoff for scenario j (j≠i) when production plan P^(i)(i.di-elect cons.S) is used for the first period, and then theproduction plan is subsequently re-optimized.

E_(i) --the expected payoff of production plan i at the beginning of theplanning horizon, i.e., ##EQU6## Δ_(j) ⁺ --the difference between themaximum payoff and the expected payoff, and

Δ_(i) ⁻ --the difference between the minimum payoff and the expectedpayoff.

3. The computation of PPPT

In general, an optimization problem can be formulated to obtain aproduction plan under a certain criterion. Let the objective function bef(P|D,w) where P is the decision variable (the production plan), D isthe demand which is a random variable, and w represents all otherparameters that affect the objective function (e.g., costs, supplyconstraints, etc.). For the simple recourse case, the optimizationproblem is given by ##EQU7##

The solution to (1) can be obtained by either an optimization solver ora heuristic-based approach.

The PPPT is a heuristic approach for solving Problem (1) in amulti-scenario setting. In the payoff table computation, each element ofthe payoff table presents the performance measure corresponding to aproduction plan in a particular demand scenario.

3.1. Diagonal elements

For the computation of diagonal element R_(i),i, i=1, . . . , N, we haveD=D^(i). The solution can be obtained by solving the following problem.##EQU8##

The solution to (2) is called Scenario solution i, which is denoted byP^(i).

3.2. Off-diagonal elements

For the off-diagonal elements, the problem becomes a constrainedoptimization problem. In the case of the canonical demand scenario tree,the general formulation for the computation of off-diagonal elementR_(ij), i≠j, can be presented as follows. ##EQU9## where P₁ is the firstcolumn of P, and P₁ ^(i) is the first column of P^(i). In this case, theassumption made for computing the off-diagonal elements of the payofftable is that the initial production plan is made based on demandscenario i but the actual demand scenario turns out to be j. Thedecision maker can adjust the production plan at the beginning of thesecond period. However, the production plan made according to scenario iis already implemented for the first period. Therefore, the decisionvariables of the first period have to be fixed in the re-optimizationwhich is based on the new scenario j.

3.2.1. Nonanticipativity

One important concept in the implementation of PPPT computation is thenonanticipativity of the production plan. The nonanticipativityassumption guarantees that the decisions made in any given period areimplementable, i.e., they do not depend on information that is not yetavailable. If a plan is nonanticipative, the decisions made in a periodare identical for any two scenarios that are identical up to thatperiod. This means that if a node is common to two different demandscenarios, the decisions must be the same at the common node for the twoproduction plans made based on the two demand scenarios. The computationof off-diagonal elements should respect the nonanticipative assumptionin order to make the production plan implementable. One such example isillustrated in FIG. 2 (numeral 14), where the nonanticipavity requiresthat P¹ ₁ =P² ₁. In the case of canonical scenario trees, thenonanticipativity is implied in the formulation shown in (3) since theonly common node is the root and the re-optimization always takes placein the second period.

In general, the requirements for the nonanticipative assumption can bewritten as follows.

    P.sub.t.sup.i =P.sub.t.sup.j, t=1, . . . , τ, ∀i,j that are identical up to τ.                                    (4)

However, the computation of off-diagonal elements when the scenario treeis non-canonical form is not as straightforward as for canonicalscenario trees. The difficulty is that for every common node, thenonanticipativity requires the decisions at the node to be the same forall demand scenarios sharing the node. The solution respecting such aproperty, i.e., condition (4), and at the same time without compromisingthe optimality would require the use of stochastic LP technique, whichcould be computationally complex. To overcome this difficulty, aheuristic is adapted in the PPPT computation for the scenario tree withnon-canonical form.

3.2.2. The Algorithm for Computing R_(ij)

Without loss of generality, we assume p₁ ≧p₂ ≧. . . ≧p_(N), where p_(j)is the probability of scenario j.

Diagonal elements R_(i),i is computed the same way as in (2).

For off-diagonal element R_(ij), i≠j, if a node of scenario j is commonto scenario i in period n, let

    P.sub.n =P.sub.n.sup.i.                                    (5)

If a node of scenario j is common to any scenarios other than i, let i'be the smallest index of all these scenarios. If i'<j, let

    P.sub.n =P.sub.n.sup.i'.                                   (6)

Off-diagonal element R_(ij) is then obtained by solving (2) withconstraints (5) and (6).

3.3. Upper and Lower Bounds

PPPT also provides the upper and lower bounds of the optimal solutionfor the stochastic programming problem with full recourse. The upper andlower bounds are given by ##EQU10## respectively.

Proof.

Denote the optimal solution by P*. The expected payoff of P* is given by##EQU11## where R(P*|D^(i)) is the payoff of P* under scenario i.

Since R_(ii) ≧R(P*|D^(j)), ##EQU12##

On the other hand, since P* is optimal, its expected payoff is at leastas good as the expected payoff of any scenario solution, i.e.,

R*≧E_(i), for all i.

3.4. Optimization Engine

In the PPPT implementation, we preferably use SCE as the optimizationengine. SCE is a production planning optimization software developed atIBM Research for computing the capability to supply finished goods basedon availability of constrained components. SCE is based on the implosiontechnology. It can perform resource allocation under constraints byusing demands, available resources, and the Bill Of Manufacture(includes BOM as well as Bill of Capacities) to determine a feasibleproduct mix which meets the user defined criterion. The type of theobjective function used by SCE can be one of the three options: Revenue,Profit, or Priority. Among them, Priority is not used in the PPPTcomputation. Furthermore, since SCE does not include cost informationfor inventory holding, backlog penalty, and obsolescence, the profitobtained by SCE will be adjusted to reflect these costs. However, thesecosts are computed after the SCE optimization is completed.

The diagonal elements of the PPPT are obtained by running SCE for thegiven reference scenarios. The off-diagonal elements are computed byrunning SCE with the demand given by the new scenario and the productionconstraints imposed by the production plan made based on the referencescenario and the nonanticipative assumption. For example, for anoff-diagonal element which represents the performance measure underscenario j for the production plan made initially based on scenario i,we first obtain the production constraints (5) and (6), then run SCEagainst the demand scenario j.

4. The Graphical User Interface

The graphical user interface is built in forms of World Wide Web (WWW)pages. The programs implementing the PPPT computation are installed on aserver which is also the Web server hosting the WWW pages for thegraphical user interface of PPPT. All the required data are stored onthe same server. A user accesses the graphical user interface of PPPT bylinking a Web browser to the Universal Resource Locator (URL) of theserver. A Logon page will be presented when the connection isestablished (See FIGS. 3-8, numerals 16-26).

The Logon Page (see FIG. 3)

The user is required to enter a valid pair of userid and password. Ifthe userid and the password entered are not valid, further access toother PPPT WWW pages will be denied. Otherwise, the Web browser willconnect to the PPPT Main Page.

The Main Page (see FIG. 4)

A Main Task List table will be presented. The four major steps of thePPPT computation are listed with a brief description for each step. Thecurrent status of each of the four steps is also reported in the table.The user should choose an activity from the Main Task List.

set parameters (see FIG. 5): this step allows the user to view/changethe current setting of the following parameters: the number of demandscenarios, the number of periods in the planning horizon, the type ofoptimization engine to be used, and the type of objective foroptimization.

modify data: allows the user to view/modify the data used for the PPPTcomputation. There are four types of data files to be viewed/modified:

Scenario File,

Bill of Materials File,

Supply Volume File, and

Demand Volume Files

The user can choose one of the files for viewing/editing.

compute payoff table: invokes the server programs to perform the desiredPPPT computation.

Upon the completion of the PPPT computation, the message "Pay-off Tablecomputation is completed!" will be displayed.

view payoff table: allows the user to view the payoff table in eithertable format or chart format.

Data Viewing/Editing Pages

For the Bill of Materials file, no editing capability is provided. ForScenario, Supply Volume, and Demand Volume files, the user can view andedit the data if desired (see FIG. 6). A complete table will bepresented first for viewing. If editing is allowed, the user can clickon the line number to enter the editing mode. Only one row will bedisplayed at a time in the editing mode.

PPPT Display Pages

If the table format is selected, a user may choose one performancemeasure to be displayed from the following three choices: Revenue,Profit, and Serviceability. If Profit is selected, the user may alsoprovide the backlog penalty factor and the obsolescence factor asrequired for the profit computation. The payoff table displayed in thetable form contains the complete payoff table of the selectedperformance measure and the weighted average performance measure foreach plan as well as the differences between the weighted average andthe best(worst) performance measure of the plan against a particularscenario. The plan with the best weighted average performance measurewill be highlighted in the table (see FIG. 7).

In the bar chart format, a user may choose to display a bar chart thatis corresponding to a row or a column in the payoff table, i.e., theperformance measures of a given plan against different scenarios or theperformance measures of different plans for a given scenario. Theperformance measure displayed in the bar chart form can be eitherRevenue or Profit or Serviceability. The backlog penalty factor and theobsolescence factor are required as inputs when Profit is selected (seeFIG. 8).

5. An Example

The invention is now referenced by an illustrative example. For machinerealization of the invention, one may consider the example parameters inthe following Tables II-X to be inputs for operation thereupon by themethod programmed in Perl and effectuated by a CPU and memory, and TableXI or FIGS. 7, 8 to be illustrative output displays.

Description

This is a two-period problem with six products and four demandscenarios.

Data preparation

The data required for the PPPT computation are listed in Tables 2-10.

                  TABLE 2                                                         ______________________________________                                        Demand Volume File 1                                                          Part Number                                                                              Geography     Period 1                                                                              Period 2                                     ______________________________________                                        SUP-DT     WW            1,865   1,892                                        MC-DT      WW            12,450  15,040                                       SUP-NB     WW            10,300  8,930                                        MC-NB      WW            6,700   8,500                                        SUP-SVR    WW            7,540   7,990                                        MC-SVR     WW            5,200   6,400                                        ______________________________________                                    

                  TABLE 3                                                         ______________________________________                                        Demand Volume File 2                                                          Part Number                                                                              Geography     Period 1                                                                              Period 2                                     ______________________________________                                        SUP-DT     WW            17,718  17,974                                       MC-DT      WW            11,828  14,288                                       SUP-NB     WW            12,360  10,716                                       MC-NB      WW             8,040  10,200                                       SUP-SVR    WW             7,540   7,990                                       MC-SVR     WW             5,200   6,400                                       ______________________________________                                    

                  TABLE 4                                                         ______________________________________                                        Demand Volume File 3                                                          Part Number                                                                              Geography     Period 1                                                                              Period 2                                     ______________________________________                                        SUP-DT     WW            21,448  21,758                                       MC-DT      WW            14,318  17,296                                       SUP-NB     WW            11,845  10,270                                       MC-NB      WW             7,705   9,775                                       SUP-SVR    WW             8,671   9,189                                       MC-SVR     WW             5,980   7,360                                       ______________________________________                                    

                  TABLE 5                                                         ______________________________________                                        Demand Volume File 4                                                          Part Number                                                                              Geography     Period 1                                                                              Period 2                                     ______________________________________                                        SUP-DT     WW            20,375   2,067                                       MC-DT      WW            13,602  16,431                                       SUP-NB     WW            14,214  12,323                                       MC-NB      WW             9,246  11,730                                       SUP-SVR    WW             8,671   9,189                                       MC-SVR     WW             5,980   7,360                                       ______________________________________                                    

                  TABLE 6                                                         ______________________________________                                        Supply Volume File                                                            Part Number                                                                              Geography     Period 1                                                                              Period 2                                     ______________________________________                                        MEM-4MB    WW            250,000 250,000                                      ______________________________________                                    

                  TABLE 7                                                         ______________________________________                                        Bill of Materials File                                                        Parent Part                                                                             Child Part                                                          Number    Number       Geography Usage Rate                                   ______________________________________                                        SUP-DT    P-486        WW        1                                            SUP-DT    HD-240       WW        1                                            SUP-DT    MEM-4MB      WW        1                                            MC-DT     P-486        WW        1                                            MC-DT     HD-480       WW        1                                            MC-DT     MEM-4MB      WW        1                                            SUP-NB    P-PENTIUM    WW        1                                            SUP-NB    HD-480       WW        1                                            SUP-NB    MEM-4MB      WW        2                                            SUP-NB    CD-ROM       WW        1                                            MC-NB     P-PENTIUM    WW        1                                            MC-NB     HD-720       WW        1                                            MC-NB     MEM-4MB      WW        2                                            MC-NB     CD-ROM       WW        1                                            SUP-SVR   P-POWERPC    WW        1                                            SUP-SVR   HD-720       WW        1                                            SUP-SVR   MEM-4MB      WW        4                                            SUP-SVR   CD-ROM       WW        1                                            SUP-SVR   TOK-RING     WW        1                                            MC-SVR    P-POWERPC    WW        1                                            MC-SVR    HD-720       WW        1                                            MC-SVR    MEM-4MB      WW        4                                            MC-SVR    CD-ROM       WW        1                                            MC-SVR    MULT-MED     WW        1                                            ______________________________________                                    

                  TABLE 8                                                         ______________________________________                                        Scenario File                                                                 Scenario  Period 1    Period 2                                                                              Probability                                     ______________________________________                                        1         1           1       0.42                                            2         2           2       0.18                                            3         3           3       0.28                                            4         4           4       0.12                                            ______________________________________                                    

                  TABLE 9                                                         ______________________________________                                        Revenue File                                                                  Part Number    Geography Revenue                                              ______________________________________                                        SUP-DT         WW        1,000                                                MC-DT          WW        1,100                                                SUP-NB         WW        2,000                                                MC-NB          WW        2,400                                                SUP-SVR        WW        3,500                                                MC-SVR         WW        4,000                                                ______________________________________                                    

                  TABLE 10                                                        ______________________________________                                        Profit File                                                                   Part Number    Geography Profit                                               ______________________________________                                        SUP-DT         WW        250                                                  MC-DT          WW        250                                                  SUP-NB         WW        500                                                  MC-NB          WW        700                                                  SUP-SVR        WW        600                                                  MC-SVR         WW        1,100                                                ______________________________________                                    

Procedure

1. Start a Web Browser and link to the URL of the PPPT Web server.

2. When prompted, enter the userid and password.

3. On the Main Task List page, select "set parameters".

4. On the Set Parameters page, enter "4" for the number of scenarios,"2" for the number of periods, select "LP Optimization" for theoptimization engine, and "Profit" for the objective type. Then click the"Submit" button.

5. Go back to the Main Task List page. Select "modify data".

6. Modify the data by following appropriate links as desired.

7. Go back to the Main Task List page. Select "compute payoff table".

8. Wait until a screen with the message "The Payoff Table computation iscompleted".

9. Go to the View Payoff Table Results page, select "view output tables"to view the payoff table in table format, or select "view output charts"to view the payoff table in bar-chart format.

10. On the View Output Tables page, select one from "Profit", "Revenue",and "Serviceability". If "profit" is selected, enter the values for"Backlog penalty" and "Obsolescence factor". Then click on "Submit" toview the output.

11. On the View Output charts page, select either "Plan" or "Scenario"and the number, also select one from "Profit", "Revenue", and"Serviceability". If "profit" is selected, enter the values for "Backlogpenalty" and "Obsolescence factor". Then click on "Submit" to view theoutput.

12. Repeat any step(s) as desired.

Outputs

The results of this example are summarized in Table 11. The revenue andprofit figures are in million dollars.

                  TABLE 11                                                        ______________________________________                                        PPPT Results                                                                  j      scenario 1      2    3    4    Statistics                              ρ.sub.j                                                                          probability                                                                            0.42   0.18 0.28 0.12 Mean Δ+                                                                           Δ-                      ______________________________________                                        P.sup.1                                                                              revenue  243.5  240.1                                                                              243.5                                                                              243.5                                                                              242.9                                                                              1.4  -2.8                                 profit   41.57  35.77                                                                              30.77                                                                              26.79                                                                              35.73                                                                              5.84 -8.94                         P.sup.2                                                                              revenue  240.3  255.1                                                                              251.5                                                                              255.1                                                                              247.8                                                                              7.3  -7.5                                 profit   35.87  43.66                                                                              33.99                                                                              32.35                                                                              36.32                                                                              7.34 -3.97                         P.sup.3                                                                              revenue  243.5  251.7                                                                              280  276.1                                                                              259.1                                                                              20.9 -16.6                                profit   30.89  34.23                                                                              47.8 41.14                                                                              37.46                                                                              10.34                                                                              -6.57                         P.sup.4                                                                              revenue  243.5  255.1                                                                              276.3                                                                              293.4                                                                              260.8                                                                              32.6 -17.3                                profit   26.99  32.48                                                                              41.26                                                                              50.21                                                                              34.76                                                                              15.45                                                                              -7.77                         ______________________________________                                    

Backlog penalty factor=0.2, obsolescence factor=0.5

In this PPPT, both the revenues and the profits are listed for thecomparison purpose. The highest mean in terms of revenues is archived byPlan 4, while the highest mean in terms of profits is archived by Plan3. Furthermore, if one's objective is to minimize the variability of theperformance under different scenarios, the best plan will be the onewith the smallest Δ+ and Δ- (Plan 1 in this example).

We can also obtain the upper and lower bounds of the optimal solutionfrom the table.

Revenue:

R_(U) =261.8 million dollars, R_(L) =260.8 million dollars.

Profit:

R_(U) =44.72 million dollars, R_(L) =37.46 million dollars.

What is claimed:
 1. A method for producing planning in an uncertaindemand environment, said method comprising the steps of:a. providingplurality of demand scenarios having discrete time periods, and aprobability for each of said demand scenarios; b. representinguncertainty in a demand environment by employing a scenario-basedanalysis including the steps of performing multiple optimization runsagainst different demand scenarios; and c. generating optimaldeterministic solutions for each of said demand scenarios using animplosion technology; d. computing expected payoffs for said demandscenarios and choosing an optimal production plan having a best of saidexpected payoffs; and e. modifying manufacturing of said productsaccording to said chosen optimal production plan.
 2. The method of claim1, wherein said generating step comprises the following steps:1.generating production plans for each of said demand scenarios having afixed production decision
 2. generating production plans for each ofsaid demand scenarios having a varied production decision using saidproduction plan generated in step (b) for a first discrete time periodand solving for demand scenarios for remaining discrete time periods asindicated in said production decision using an implosion technology. 3.The method of claim 2, wherein said optimal production plan is createdbased initially on a particular demand scenario.
 4. The method of claim3, wherein said optimal production is adjusted according to unfoldingsequel demand scenarios.
 5. The method of claim 4, wherein said optimalproduction is obtained using an implementable production policy for anentire demand scenario tree.
 6. The method of claim 5, wherein anexpected performance measure is computed for each said implementableproduction policy based on a particular initial demand scenario.
 7. Acomputer program device readable by a machine, tangibly embodying aprogram of instructions executable by a machine to perform method stepsfor production planning in an uncertain demand environment, said methodcomprising the steps of:a. providing plurality of demand scenarioshaving discrete time periods, and a probability for each of said demandscenarios; b. representing uncertainty in a demand environment byemploying a scenario-based analysis including the steps of performingmultiple optimization runs against different demand scenarios; and c.generating optimal deterministic solutions for each of said demandscenarios using an implosion technology; d. computing expected payoffsfor said demand scenarios and choosing an optimal production plan havinga best of said expected payoffs; and e. modifying manufacturing of saidproducts according to said chosen optimal production plan.
 8. The deviceof claim 7, wherein said generating step comprises the followingsteps:
 1. generating production plans for each of said demand scenarioshaving a fixed production decision2. generating production plans foreach of said demand scenarios having a varied production decision usingsaid production plan generated in step (b) for a first discrete timeperiod and solving for demand scenarios for remaining discrete timeperiods as indicated in said production decision using an implosiontechnology.
 9. The device of claim 8, wherein said optimal productionplan is created based initially on a particular demand scenario.
 10. Thedevice of claim 9, wherein said optimal production is adjusted accordingto unfolding sequel demand scenarios.
 11. The device of claim 10,wherein said optimal production is obtained using an implementableproduction policy for an entire demand scenario tree.
 12. The device ofclaim 11, wherein an expected performance measure is computed for eachsaid implementable production policy based on a particular initialdemand scenario.